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Chapter 2: Problem 124

The principal argument of \(-2 \sqrt{3}-2 \mathrm{i}\) is (a) \([(-5 \pi) / 6]\) (b) \([(5 \pi) / 6]\) (c) \([(-2 \pi) / 3]\) (d) \([(2 \pi) / 3]\)

Short Answer

Expert verified
The principal argument of the given complex number is \(-\frac{2 \pi}{3}\), which corresponds to option (c).

Step by step solution

01

Express the given complex number in rectangular form

The given complex number is -2√3 - 2i.
02

Identify the real and imaginary parts of the complex number

In the given complex number -2√3 - 2i, the real part is -2√3, and the imaginary part is -2.
03

Calculate the angle (theta) using the arctan function

The formula to find the angle of the complex number is: θ = arctan(Im(z)/Re(z)) Where Im(z) is the imaginary part, and Re(z) is the real part of the complex number. Θ = arctan((-2)/(-2√3))
04

Evaluate the arctan function

Upon evaluating the arctan function, we get: θ = arctan(1/√3) Since we want to find the principal argument of the complex number, ensure the angle lies between -π and π: θ = - π/3 ≈ -1.047
05

Find the corresponding option in terms of π

Since θ = - π/3 ≈ -1.047 , let's check which option matches this angle: (a) - (5π) / 6 (b) (5π) / 6 (c) - (2π) / 3 (d) (2π) / 3 The correct option is (c) - (2π) / 3 because it matches the calculated angle of θ = - π/3.

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Most popular questions from this chapter

It \(\mathrm{z}^{2}+\mathrm{z}+1=0\) where \(\mathrm{z}\) is a complex number, then the value of \([z+(1 / z)]^{2}+\left[z^{2}+\left(1 / z^{2}\right)\right]^{2}+\left[z^{3}+\left(1 / z^{3}\right)\right]^{2}+\ldots\) \(+\left[z^{6}+\left(1 / z^{6}\right)\right]^{2}\) is (a) 18 (b) 54 (c) 6 (d) 12

If \(\mathrm{x}=\mathrm{a}+\mathrm{b}, \mathrm{y}=\mathrm{a} \alpha+\mathrm{b} \beta\) and \(\mathrm{z}=\mathrm{a} \beta+\mathrm{b} \alpha\), Where \(\alpha, \beta \neq 1\) are cube roots of unity, then \(\mathrm{xyz}=\) (a) \(2\left(\mathrm{a}^{3}+\mathrm{b}^{3}\right)\) (b) \(2\left(a^{3}-b^{3}\right)\) (c) \(\mathrm{a}^{3}+\mathrm{b}^{3}\) (d) \(a^{3}-b^{3}\)

If \(z\) is a complex number satisfying \(|z-\operatorname{iRe}(z)|=|z-\operatorname{Im}(z)|\) then \(\mathrm{z}\) lies on (a) \(\mathrm{y}=\mathrm{x}-1\) (b) \(\mathrm{y}=\pm \mathrm{x}\) (c) \(\mathrm{y}=\mathrm{x}+1\) (d) \(\mathrm{y}=-\mathrm{x}+1\)

If \([(1+2 i) /(2+i)]=r(\cos \theta+i \sin \theta)\), then (a) \(r=1, \theta=\tan ^{-1}(4 / 3)\) (b) \(\mathrm{r}=\sqrt{5}, \theta=\tan ^{-1}(5 / 4)\) (c) \(\mathrm{r}=1, \theta=\tan ^{-1}(3 / 4)\) (d) \(r=2, \theta=\tan ^{-1}(3 / 4)\)

If \(z=x+i y, x, y \in R\) and \(3 x+(3 x-y) i=4-6 i\) then \(z=\) (a) \((4 / 3)+\mathrm{i} 10\) (b) \((4 / 3)-\mathrm{i} 10\) (c) \([(-4) / 3]+\mathrm{i} 10\) (d) \([(-4) / 3]-\mathrm{i} 10\)
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